Timeline for Faithful representation into $\operatorname{GL}(9,3)$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 23, 2020 at 9:24 | comment | added | Geoff Robinson | @Derek : You are right, that was an aberration on my part. Of course, as soon as the minimum polynomial of a unipotent element gets above p, the element has order at least $p^2, $ which is where I was daydreaming. | |
| Oct 23, 2020 at 7:45 | comment | added | Derek Holt | @GeoffRobinson That's not true - $C_3 \wr C_3$ embeds into ${\rm GL}(4,3)$. But I have checked by computer that no embedding into ${\rm GL}(n,3)$ for $n=4$ or 5 has the required property. I may be able to check $n=6$ - I am not sure yet - but I will not be able to get up to $n=9$ by naive computation. Still, I am starting to think that no such embedding is possible. | |
| Oct 23, 2020 at 2:58 | comment | added | HIMANSHU | @YCor Any $ n \times n$ matrix, $n \geq 9$ will serve. | |
| Oct 22, 2020 at 15:09 | comment | added | YCor | Any particular reason to ask about $9\times 9$ matrices? | |
| Oct 22, 2020 at 15:08 | comment | added | YCor | Reversing, it amounts to ask whether there's a 4-dimensional subalgebra of the (non-unital) algebra of strictly upper triangular $9\times 9$ matrices over $\mathbf{F}_3$ on which the law $(a,b)\mapsto a+b+ab$ defines a group law isomorphic to $C_3\wr C_3$. | |
| Oct 22, 2020 at 7:56 | comment | added | HIMANSHU | @DerekHolt Yes, exactly. | |
| Oct 22, 2020 at 7:53 | comment | added | Derek Holt | We can assume that the image lies in a given Sylow $3$-subgroup of ${\rm GL}(9,3)$, so you are asking whether there is a subgroup $S$ of the group of upper unitriangular matrices in ${\rm GL}(9,3)$ with $S \cong C_3 \wr C_3$, such that $\{ g - I : g \in S \}$ is a group under addition. | |
| Oct 22, 2020 at 7:42 | comment | added | HIMANSHU | Actually it started with $C_3 \wr C_3$, it can be embedded in $S_9$, as the set $T$ taken above. If I will be able to find this kind of mapping into $GL(9,3)$, then I can say many things about the group ring $F_3(C_3 \wr C_3)$ . | |
| Oct 22, 2020 at 7:33 | comment | added | Derek Holt | Could you possibly provide some kind of motivation for this question? | |
| Oct 22, 2020 at 6:43 | history | edited | YCor | CC BY-SA 4.0 |
increased parentheses for readibility
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| Oct 22, 2020 at 6:33 | history | asked | HIMANSHU | CC BY-SA 4.0 |